Due to the simple form of the equation, it has been used as a joke.

Bremner-Macleod numbers are the smallest positive integer solutions \((a,b,c)\) of the following Diophantine equation:

$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=N$$ where \(N\) is a positive integer. The equation and the sizes of its solutions were discussed by Andrew Bremner and Allan Macleod.[1]

Size

For all odd \(N\) and some even \(N\), the equation has no positive integer solutions.

The Bremner-Macleod number for \(N=4\) is (a,b,c) =

(154476802108746166441951315019919837485664325669565431700026634898253202035277999,

36875131794129999827197811565225474825492979968971970996283137471637224634055579,

4373612677928697257861252602371390152816537558161613618621437993378423467772036)

For \(N=6\) it is

(20260869859883222379931520298326390700152988332214525711323500132179943287700005601210288797153868533207131302477269470450828233936557,

2250324022012683866886426461942494811141200084921223218461967377588564477616220767789632257358521952443049813799712386367623925971447,

1218343242702905855792264237868803223073090298310121297526752830558323845503910071851999217959704024280699759290559009162035102974023)

For \(N=10\) it is

(4862378745380642626737318101484977637219057323564658907686653339599714454790559130946320953938197181210525554039710122136086190642013402927952831079021210585653078786813279351784906397934209,

269103113846520710198086599018316928810831097261381335767926880507079911347095440987749703663156874995907158014866846058485318408629957749519665987782327830143454337518378955846463785600977,

221855981602380704196804518854316541759883857932028285581812549404634844243737502744011549757448453135493556098964216532950604590733853450272184987603430882682754171300742698179931849310347)

For \(N\le200\), the amounts of digits in \(\max\{a,b,c\}\) are shown below:

N # digits
4 81
6 134
10 190
12 2707
14 1876
16 414
18 10323
24 33644
28 81853
32 14836
38 1584369
42 886344
46 198771
48 418086
58 244860
60 9188
66 215532
76 23662
82 85465
92 252817
102 625533
112 935970
116 112519
126 196670
130 8572242
132 3607937
136 26942
146 259164
156 12046628
158 15097279
162 1265063
178 398605460
182 2828781
184 20770896
186 5442988
196 11323026
198 726373
200 71225279

For \(N=896\), \(a\) has more than 2.187 trillion digits.

Sources

  1. Andrew Bremner and Allan Macleod, An unusual cubic representation problem
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